Finite Group Databasebeta

← #44561889

Subgroups — #44561889

Order 32128.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548742C211yesmaximal, characteristic, complemented
2C2×#22548738C211yesmaximal, characteristic, complemented
3#30632340C211yesmaximal, characteristic, complemented
4#30632330C211yesmaximal, characteristic, complemented
5#30631942C211yesmaximal, characteristic, complemented
6#30631913C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631889C211yesmaximal, characteristic, complemented
9#30631848C211yesmaximal, characteristic, complemented
10#30631848C211yesmaximal, characteristic, complemented
11#30631836C211yesmaximal, characteristic, complemented
12#30631831C211yesmaximal, characteristic, complemented
13#30631800C211yesmaximal, characteristic, complemented
14#30631760C211yesmaximal, characteristic, complemented
15#497×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860811C2211yescharacteristic
18C2×#16860811C2211yescharacteristic, complemented
19C2×D4×D251C2211yescharacteristic
20#22548742—24no—
21#22548738—24no—
22C2×#16860797C2211yescharacteristic
23C2×#16860797C2211yescharacteristic
24C2×#16860797C2211yescharacteristic, complemented
25C2×D2008C2211yescharacteristic
26C2×#16860795C2211yescharacteristic
27C2×#16860793C2211yescharacteristic
28C2×C8×D251C2211yescharacteristic, complemented
29#22548700—21no—
30#22548700—21no—
31#22548697—21no—
32#22548691C2211yescharacteristic
33#22548691C2211yescharacteristic
34#22548691C2211yescharacteristic, complemented
35#22548690—21no—
36#22548688—21no—
37#22548686C2211yescharacteristic
38#22548685—21no—
39C4×D1004C2211yescharacteristic
40C4×D1004C2211yescharacteristic
41C4×D1004C2211yescharacteristic, complemented
42C4⋊C4×D251—21no—
43C22⋊C4×D251—21no—
44#22548647C2211yescharacteristic, complemented
45C4⋊(C251⋊C8)C2211yescharacteristic, complemented
46#22548639C2211yescharacteristic, complemented
47#22548639C2211yescharacteristic, complemented
48#22548639C2211yescharacteristic, complemented
49#22548637C2211yescharacteristic
50#22548637C2211yescharacteristic, complemented
51#22548636C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53(C2×D251)⋊C8C2211yescharacteristic, complemented
54C2×Q16×C251C2211yescharacteristic, complemented
55C2×Q16×C251C2211yescharacteristic, complemented
56C2×SD16×C251C2211yescharacteristic
57C2×SD16×C251C2211yescharacteristic
58C4⋊Q8×C251C2211yescharacteristic, complemented
59#124×C251C2211yescharacteristic
60#97×C251C2211yescharacteristic, complemented
61C23×D251—21no—
62#16860811—22no—
63#16860811—22no—
64#16860811—22no—
65#16860811—41no—
66#16860811—41no—
67#16860811—41no—
68C2×D1004—21no—
69C2×D1004—21no—
70C2×D1004—21no—
71C2×D1004D411yescharacteristic
72C2×D1004C2311yescharacteristic
73C2×D1004C2311yescharacteristic
74C2×C22⋊D251—21no—
75C2×C22⋊D251—21no—
76D4×D251—22no—
77D4×D251—41no—
78C2×C4×D251—21no—
79C2×C4×D251—21no—
80C2×C4×D251—21no—
81C2×C4×D251—21no—
82C2×C4×D251—21no—
83C2×C4×D251C2311yescharacteristic
84C2×C4×D251C2311yescharacteristic
85#16860797—22no—
86#16860797—22no—
87#16860797—22no—
88D2008D414yescomplemented
89#16860795—22no—
90#16860793—22no—
91C2×C251⋊C8C2311yescharacteristic
92C2×C251⋊C8C2311yescharacteristic
93C8×D251—22no—
94C4⋊Dic251D411yescharacteristic
95C4⋊Dic251C2311yescharacteristic
96C4⋊Dic251C2311yescharacteristic
97Dic251⋊C4—21no—
98Dic251⋊C4—21no—
99Dic251⋊C4—21no—
100C22⋊Dic251—21no—
101(C2×D251)⋊C4—21no—
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4—21no—
105(C2×D251)⋊C4—21no—
106(C2×D251)⋊C4D411yescharacteristic
107(C2×D251)⋊C4D411yescharacteristic
108C2×Q8×C251C2311yescharacteristic
109C2×Q8×C251C2311yescharacteristic
110C2×Q8×C251C2311yescharacteristic
111C2×D4×C251C2311yescharacteristic
112Q16×C251—22no—
113Q16×C251—41no—
114SD16×C251—22no—
115SD16×C251—41no—
116C2×C8×C251C2311yescharacteristic
117C2×C8×C251C2311yescharacteristic
118C4⋊C4×C251—21no—
119C4⋊C4×C251—21no—
120C22⋊C4×C251—21no—
121C22⋊C4×C251—21no—
122C42×C251C2311yescharacteristic
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—42no—
126C22×D251—42no—
127C22×D251—81no—
128C22×D251C2×D411yescharacteristic
129C22×D251C2×D411yescharacteristic
130D1004—21no—
131D1004—21no—
132D1004—41no—
133D1004—42no—
134D1004—42no—
135D1004—42no—
136D1004C2×D412yes—
137D1004C2×D412yes—
138C22⋊D251—42no—
139C22⋊D251—81no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251C2×D411yescharacteristic
143C2×Dic251C2×D411yescharacteristic
144C4×D251—21no—
145C4×D251—21no—
146C4×D251—21no—
147C4×D251—21no—
148C4×D251—42no—
149C4×D251—42no—
150C4×D251—42no—
151C4×D251—42no—
152C4×D251—81no—
153C251⋊C8—21no—
154C251⋊C8—21no—
155C23×C251—21no—
156Q8×C251—21no—
157Q8×C251—21no—
158Q8×C251—21no—
159Q8×C251—21no—
160Q8×C251—21no—
161Q8×C251—21no—
162D4×C251—21no—
163D4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251C2×D411yescharacteristic
168C2×C4×C251C2×D411yescharacteristic
169C2×C4×C251C2411yescharacteristic
170C8×C251—21no—
171C8×C251C2×D412yes—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251C22×D411yescharacteristic
176C2×D251—21no—
177C2×D251—21no—
178C2×D251—22no—
179C2×D251—22no—
180C2×D251—21no—
181C2×D251—21no—
182C2×D251—21no—
183C2×D251—21no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—84no—
189C4×C251—21no—
190C4×C251—21no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251—41no—
194C4×C251C22×D411yescharacteristic
195C4×C251H(5,𝔽2)11yescharacteristic
196Dic251—21no—
197Dic251—21no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—81no—
201C2×C251D4211yescharacteristic
202C2×C251#57511yescharacteristic
203C2×C251#57711yescharacteristic
204D251—41no—
205D251—41no—
206D251—42no—
207D251—42no—
208C251#326711yescharacteristic, complemented, Sylow
209#3267—2511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2511no—
211C2×#135—2511no—
212#547—2511no—
213#546—2511no—
214#497D25111yescharacteristic, complemented
215#470—2511no—
216#465—2511no—
217#463—2511no—
218#461—2511no—
219#452—2511no—
220#452—2511no—
221#450—2511no—
222#449—2511no—
223#442—2511no—
224#435—2511no—
225C2×#41—2511no—
226C2×#41—2511no—
227C2×#41—2511no—
228C22×D4—2511no—
229AGL(1,ℤ/8)—5024no—
230#135—5024no—
231C2×Q16C2×D25111yescharacteristic, complemented
232C2×Q16C2×D25111yescharacteristic, complemented
233C2×SD16—2511no—
234C2×SD16—2511no—
235C2×SD16—2511no—
236C2×SD16C2×D25111yescharacteristic
237C2×SD16C2×D25111yescharacteristic
238C2×D8—2511no—
239C2×D8—2511no—
240C2×M16—2511no—
241C22×C8—2511no—
242C4⋊Q8C2×D25111yescharacteristic, complemented
243#124C2×D25111yescharacteristic
244#123—5021no—
245#123—5021no—
246#123—5021no—
247C22⋊Q8—2511no—
248C22⋊Q8—2511no—
249C22⋊Q8—2511no—
250#121—5021no—
251#121—5021no—
252#121—2511no—
253#120—5021no—
254C4×D4—2511no—
255C4×D4—2511no—
256C4×D4—2511no—
257C2×C4⋊C4—5021no—
258C2×C22⋊C4—5021no—
259#106—2511no—
260C4⋊C8—2511no—
261#103—2511no—
262#103—2511no—
263#103—2511no—
264#102—2511no—
265#102—2511no—
266#102—2511no—
267C22⋊C8—2511no—
268C22⋊C8—2511no—
269#97C2×D25111yescharacteristic, complemented
270C24—5021no—
271#41—10041no—
272#41—10041no—
273#41—10041no—
274#41—5022no—
275#41—5022no—
276#41—5022no—
277C2×Q8C22×D25111yescharacteristic
278C2×Q8C22×D25111yescharacteristic
279C2×Q8C22×D25111yescharacteristic
280C2×D4—10041no—
281C2×D4—5021no—
282C2×D4—5021no—
283C2×D4—5021no—
284C2×D4—5022no—
285C2×D4—5021no—
286C2×D4—5021no—
287C2×D4—2511no—
288C2×D4—2511no—
289C2×D4—2511no—
290C2×D4C22×D25111yescharacteristic
291C22×C4—5021no—
292C22×C4—5021no—
293C22×C4—5021no—
294C22×C4—5021no—
295C22×C4—5021no—
296C22×C4—2511no—
297C22×C4—2511no—
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—41no—
302SD16—5022no—
303SD16—5022no—
304SD16—5022no—
305D8—5022no—
306D8—2514no—
307M16—5022no—
308C2×C8—5022no—
309C2×C8—2511no—
310C2×C8—2511no—
311C2×C8C22×D25111yescharacteristic
312C2×C8C22×D25111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—5021no—
316C4⋊C4—5021no—
317C4⋊C4—5021no—
318C4⋊C4—2511no—
319C4⋊C4—2511no—
320C4⋊C4—2511no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—5021no—
324C22⋊C4—5021no—
325C22⋊C4—5021no—
326C22⋊C4—5021no—
327C22⋊C4—5021no—
328C22⋊C4—5021no—
329C22⋊C4—2511no—
330C22⋊C4—2511no—
331C42C22×D25111yescharacteristic
332C23—21no—
333C23—20081no—
334C23—10042no—
335C23—10042no—
336C23—5021no—
337C23—5021no—
338C23—2511no—
339C23—2511no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—20081no—
349D4—10041no—
350D4—10042no—
351D4—10042no—
352D4—10042no—
353D4—10042no—
354D4—5021no—
355D4—5021no—
356D4—2512no—
357D4—2512no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—20081no—
362C2×C4—10042no—
363C2×C4—10042no—
364C2×C4—10042no—
365C2×C4—10042no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—2511no—
373C2×C4—2511no—
374C2×C4D4×D25111yescharacteristic
375C2×C4D4×D25111yescharacteristic
376C2×C4C23×D25111yescharacteristic
377C8—21no—
378C8—5021no—
379C8—5021no—
380C8D4×D25112yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—20084no—
385C22—10041no—
386C22—10041no—
387C22—10041no—
388C22—10041no—
389C22—5021no—
390C22—5021no—
391C22—5021no—
392C22—5021no—
393C22—5022no—
394C22—5022no—
395C22—5021no—
396C22—5021no—
397C22C2×D4×D25111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—10041no—
404C4—10041no—
405C4—5021no—
406C4—5021no—
407C4C2×D4×D25111yescharacteristic
408C4#2254876011yescharacteristic
409C2—81no—
410C2—10041no—
411C2—10042no—
412C2—10042no—
413C2—10041no—
414C2#3063232211yescharacteristic
415C2#3063248611yescharacteristic
416C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#497×C251C2
socleC22×C251C22×D4
hypercenter#497D251
hypocenterC251#3267
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. C2×#22548738C2semidirect
3. #30632340C2semidirect
4. #30632330C2semidirect
5. #30631942C2semidirect
6. #30631913C2semidirect
7. #30631899C2semidirect
8. #30631889C2semidirect
9. #30631848C2semidirect
10. #30631848C2semidirect
11. #30631836C2semidirect
12. #30631831C2semidirect
13. #30631800C2semidirect
14. #30631760C2semidirect
15. #497×C251C2semidirect
18. C2×#16860811C22semidirect
24. C2×#16860797C22semidirect
28. C2×C8×D251C22semidirect
34. #22548691C22semidirect
41. C4×D1004C22semidirect
44. #22548647C22semidirect
45. C4⋊(C251⋊C8)C22semidirect
46. #22548639C22semidirect
47. #22548639C22semidirect
48. #22548639C22semidirect
50. #22548637C22semidirect
53. (C2×D251)⋊C8C22semidirect
54. C2×Q16×C251C22semidirect
55. C2×Q16×C251C22semidirect
58. C4⋊Q8×C251C22semidirect
60. #97×C251C22semidirect
88. D2008D4semidirect
208. C251#3267semidirect
214. #497D251semidirect
231. C2×Q16C2×D251semidirect
232. C2×Q16C2×D251semidirect
242. C4⋊Q8C2×D251semidirect
269. #97C2×D251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225487422
C2×#225487382
#306323402
#306323302
#306319422
#306319132
#306318992
#306318892
#306318482
#306318362
#306318312
#306318002
#306317602
#497×C2512
#3267251